Separate theoretical and experimental investigations of the effect of lattice anharmonicity on the A1g phonon dynamics in photoexcited bismuth are presented. First-principles density functional calculations show that the anharmonic contribution to the phonon period is negligible for an excitation of 1.25% or less of the valence electrons, corresponding to electronic frequency softening from 2.9 to 2.3 THz. Experiments using optical double-pump-probe excitation of coherent phonon motion clearly separate the role of lattice anharmonicity from electron-hole plasma and other dynamics, confirming that the effect of anharmonicity on the phonon period is much smaller than the observed softening.